The binary field will be symbolized as $\mathbb F_2$ in the customary manner, whilst the Steenrod algebra over this field shall be written as $\mathscr A.$ In this paper, we study Singer’s conjecture [Math. Z. \textbf{202} (1989), 493-523] for the algebraic transfers of ranks 5 and 6 in the generic families of internal degrees. The Singer algebraic transfer stands as a valuable instrument for unraveling the intricate structure of the cohomology ${\rm Ext}_{\mathscr A}^{s,k} := {\rm Ext}_{\mathscr A}^{s}(\mathbb F_2, \Sigma^{k}\mathbb F_2)$ of $\mathscr A.$ Remarkably, we have shown that the indecomposable element $y\in {\rm Ext}_{\mathscr A}^{6,44}$ is not in the image of the sixth algebraic transfer.

Let $A$ denote the Steenrod algebra at the prime 2 and let $k = \mathbb Z_2.$ A very difficult problem in algebraic topology is to determine a minimal set of $A$-generators for the polynomial ring $P_q = k[x_1, \ldots, x_q] = H^{*}(k^{q}, k)$ on $q$ generators $x_1, \ldots, x_q$ with $|x_1|= |x_2| = \cdots = |x_q| =  1.$ Equivalently, one can write down explicitly a basis for the cohit space $\pmb{Q}^{q} := k\otimes_{A} P_q$ in each non-negative degree $n.$ This subject, which has now a long history, is the content of the classical “hit problem” proposed in [Abstracts Papers Presented Am. Math. Soc.  \textbf{833} (1987), 55-89]. Furthermore, it is closely related to the $q$-th transfer homomorphism $Tr_q^{A}$ constructed by William Singer in [Math. Z. \textbf{202} (1989), 493-523]. This $Tr_q^{A}$ passes from the space of $G(q)$-coinvariant $k\otimes _{G(q)} P_A((P_q)_n^{*})$ of $\pmb{Q}^{q}$ to the $k$-cohomology group of the Steenrod algebra, ${\rm Ext}_{A}^{q, q+n}(k, k),$ where $G(q)$ denotes the general linear group of degree $q$ over the field $k,$ whereas $P_A((P_q)_n^{*})$ is the primitive part of $(P_q)^{*}_n$ under the action of $A.$ Particularly, Singer conjectured that $Tr_q^{A}$ is a monomorphism, but this remains unanswered for all $q\geq 4.$ The present paper is devoted to investigating this conjecture for rank 4 in some certain internal degrees. Specifically, by the usage of the techniques of the hit problem in four variables, we explicitly determine the structure of $k\otimes _{G(4)} P_A((P_4)_{n}^{*})$ in some generic degrees $n.$ Then, applying these results and a representation of $Tr_4^{A}$ via the lambda algebra, we show that Singer’s conjecture is true for rank $q = 4$ in respective degrees $n.$ This has contributed to the final proof of Singer’s conjecture in the rank 4 case.

In recent decades, the structure of the mod-2 cohomology of the Steenrod ring $\mathscr A$ has become a major subject of study in the field of Algebraic Topology. One of the earliest attempts to study this cohomology through the use of modular representations of the general linear groups was the groundbreaking work [Math. Z. \textbf{202} (1989), 493-523] by W.M. Singer. In that work, Singer introduced a homomorphism, commonly referred to as the “algebraic transfer,” which maps from the coinvariants of a certain representation of the general linear group to the mod-2 cohomology group of the ring $\mathscr A.$ Singer’s conjecture, in particular, which states that the algebraic transfer is a monomorphism for all homological degrees, remains a highly significant and unresolved problem in Algebraic Topology. In this research, we take a major stride toward resolving the Singer conjecture by establishing its truth for the homological degree four.

Let $\mathcal A$ be the classical, singly-graded Steenrod algebra over the prime order field $\mathbb F_2$ and let $P^{\otimes h}: = \mathbb F_2[t_1, \ldots, t_h]$ denote the polynomial algebra on $h$ generators, each of degree $1.$ Write $GL_h$ for the usual general linear group of rank $h$ over $\mathbb F_2.$ Then, $P^{\otimes h}$ is an $\mathcal A[GL_h]$-module. As is well known, for all homological degrees $h \geq 6$, the cohomology groups ${\rm Ext}_{\mathcal A}^{h, h+\bullet}(\mathbb F_2, \mathbb F_2)$ of the algebra $\mathcal A$ are still shrouded in mystery. The algebraic transfer $Tr_h^{\mathcal A}: (\mathbb F_2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}]^{*})_{\bullet}\longrightarrow {\rm Ext}_{\mathcal A}^{h, h+\bullet}(\mathbb F_2, \mathbb F_2)$ of rank $h,$ constructed by W. Singer [Math. Z. \textbf{202} (1989), 493-523], is a beneficial technique for describing the Ext groups. Singer’s conjecture about this transfer states that \textit{it is always a one-to-one map}. Despite significant effort, neither a complete proof nor a counterexample has been found to date.  The unresolved nature of the conjecture makes it an interesting topic of research in Algebraic topology in general and in  homotopy theory in particular. \medskip The objective of this paper is to investigate Singer’s conjecture, with a focus on all $h\geq 1$ in degrees $n\leq 10 = 6(2^{0}-1) + 10\cdot 2^{0}$ and for $h=6$ in the general degree $n:=n_s=6(2^{s}-1) + 10\cdot 2^{s},\, s\geq 0.$ Our methodology relies on the hit problem techniques for the polynomial algebra $P^{\otimes h}$, which allows us to investigate the Singer conjecture in the specified degrees. Our work is a continuation of the work presented by Mothebe et al. [J. Math. Res. \textbf{8} (2016), 92-100] with regard to the hit problem for $P^{\otimes 6}$ in degree $n_s$, expanding upon their results and providing novel contributions to this subject. More generally, for $h\geq 6,$ we show that the dimension of the cohit module $\mathbb F_2\otimes_{\mathcal A}P^{\otimes h}$ in degrees $2^{s+4}-h$ is equal to the order of the factor group of $GL_{h-1}$ by the Borel subgroup $B_{h-1}$ for every $s\geq h-5.$ Especially, for the Galois field $\mathbb F_{q}$ ($q$ denoting the power of a prime number), based on Hai’s recent work [C. R. Math. Acad. Sci. Paris \textbf{360} (2022), 1009-1026], we claim that the dimension of the space of the indecomposable elements of $\mathbb F_q[t_1, \ldots t_h]$ in general degree $q^{h-1}-h$ is equal to the order of the factor group of $GL_{h-1}(\mathbb F_q)$ by a subgroup of the Borel group $B_{h-1}(\mathbb F_q).$ As applications, we establish the dimension result for the cohit module $\mathbb F_2\otimes_{\mathcal A}P^{\otimes 7}$ in degrees $n_{s+5},\, s > 0.$ Simultaneously, we demonstrate that the non-zero elements $h_2^{2}g_1 = h_4Ph_2\in {\rm Ext}_{\mathcal A}^{6, 6+n_1}(\mathbb F_2, \mathbb F_2)$ and $D_2\in {\rm Ext}_{\mathcal A}^{6, 6+n_2}(\mathbb F_2, \mathbb F_2)$ do not belong to the image of the sixth Singer algebraic transfer, $Tr_6^{\mathcal A}.$ This discovery holds significant implications for Singer’s conjecture concerning algebraic transfers. We further deliberate on the correlation between these conjectures and antecedent studies, thus furnishing a comprehensive analysis of their implications.

We will look at the binary field $\mathbb F_2$.  The classical “hit problem” in algebraic topology, which is widely considered to be an important and fascinating open problem that has yet to be resolved, asks for a minimal set of generators for the polynomial algebra, $\mathcal P_m:=\mathbb F_2[x_1, x_2, \ldots, x_m]$, on $m$ variables $x_1, \ldots, x_m$, each of which has degree one, regarded as a connected unstable module over the 2-primary Steenrod algebra $\mathscr A.$ The algebra $\mathcal P_m$ is the cohomology with $\mathbb F_2$-coefficients of the product of $m$ copies of the Eilenberg-MacLan complex $K(\mathbb F_2, 1)$. Despite extensive study over the past three decades, the hit problem remains unresolved for $m\geq 5$. In this article, we develop our previous work [Commun. Korean Math. Soc. \textbf{35} (2020), 371-399] on the hit problem for the $\mathscr A$-module $\mathcal P_5$ in the generic degree $n_s = 5(2^{s}-1) + 18.2^{s}$ with $s$ an arbitrary non-negative integer. As a consequence, a localized variation  of the Kameko conjecture, which concerns the dimension of the cohit space $\mathbb F_2\otimes_{\mathscr A}\mathcal P_m$ in relation to parameter vectors, has been claimed to be veracious in the instance where $m = 5$ and the degree is $n_s.$ Also, we demonstrate that this conjecture remains valid for all $m\geq 1$ and degrees $\leq 12.$ This study has two important applications:  (1) it establishes the dimension result for the cohit space $\mathbb F_2\otimes_{\mathscr A}\mathcal P_m$ for $m = 6$ in the generic degree $5(2^{s+4}-1) + n_1.2^{s+4}$ with $s > 0;$ and (2) it describes the representations of the general linear group of rank $5$ over $\mathbb F_2.$ As a result, we prove that the algebraic transfer, defined by William Singer [Math. Z. \textbf{202} (1989), 493-523], is an isomorphism in bidegrees $(5, 5+n_s)$ with $s\geq 0.$ Besides, we obtain new results on the behavior of this algebraic transfer for all homological degrees $m$. Specifically, we show that Singer’s transfer is a trivial isomorphism in bidegree $(m, m+12)$ for any $m > 0$. At the end of this work, we discuss the hit problem for the symmetric polynomial algebra $\mathcal P_m^{\Sigma_m}.$ This topic has been previously studied by Ali Janfada and Reginald Wood for $m\leq 3.$

Let $P_s:= \mathbb F_2[x_1,x_2,\ldots ,x_s]$ be the graded polynomial algebra over the prime field of two elements, $\mathbb F_2$, in $s$ variables $x_1, x_2, \ldots , x_s$, each of degree one. This algebra is considered as a graded module over the  mod-2 Steenrod algebra, $\mathscr {A}$. The classical “hit problem”, initiated by Frank Peterson [Abstracts Amer. Math. Soc. 833 (1987), 55-89], concerned with seeking a minimal set of $\mathscr A$-module $P_s.$ Equivalently, when $\mathbb F_2$ is an $\mathscr A$-module concentrated in degree 0, one can write down explicitly a monomial basis for the $\mathbb Z$-graded vector space over $\mathbb F_2$: $$ QP_s:= \mathbb F_2 \otimes_{\mathscr A} P_s = P_s/\mathscr A^+\cdot P_s,$$ where $\mathscr A^{+}$ denotes the augmentation ideal of $\mathscr A.$ The problem is unresolved in general. In this paper, we study the hit problem for $P_s$ with $s\geq 5.$ More explicitly, we first compute explicitly the dimension of $QP_s$ for $s = 5$ in the generic degree $21\cdot 2^{t}-5$ with $t = 1.$ Note that the problem when $t = 0$ was solved by N. Sum [Vietnam J. Math. 49 (2021), 1079-1096]. Next, we study the dimension of $QP_s$ in degrees $s+5$ for $8\leq s\leq 9.$ This study corrects some results in Moetele and Mothebe’s paper [East-West Journal of Mathematics 18 (2016), 151-170]. We also give an explicit formula for the dimension of $QP_s$ in degree $14$ for all $s > 0$ and in degree $15$ for all $s > 0,\, s\neq 10.$ As applications, we investigate William Singer’s conjecture [Math. Z. 202 (1989), 493-523] on the algebraic transfer of rank $5$ in degrees $21\cdot 2^{t}-5$ for all $t\geq 0$ and of ranks $s > 0$ in internal degrees $d,\, 13\leq d\leq 15.$ This is a completely new result of proving the Singer conjecture for all ranks $s$ in certain internal degrees. In particular, our results have shown that any element in the $Sq^{0}$-families $\{\chi_t=(Sq^{0})^{t}(\chi_0)\in {\rm Ext}_{\mathscr A}^{5, 21\cdot 2^{t+1}}(\mathbb F_2, \mathbb F_2)|\, t\geq 0\}$ and $\{D_1(t)=(Sq^{0})^{t}(D_1(0))\in {\rm Ext}_{\mathscr A}^{5, 57\cdot 2^{t}}(\mathbb F_2, \mathbb F_2)|\, t\geq 0\}$ belongs to the image of the algebraic transfer of rank $5.$ For higher ranks, we explore the behavior of the algebraic transfer of rank 7 in the generic degrees $23\cdot 2^{t}-7$ for $t = 0$ and $\ell\cdot 2^{t}-7$ for $\ell\in \{9,\, 16\},\,  t\leq 3.$ Our results then claim that the non-zero elements $Pc_0\in {\rm Ext}_{\mathscr A}^{7, 23\cdot 2^{0}}(\mathbb F_2, \mathbb F_2),$\ $k_0 = k\in {\rm Ext}_{\mathscr A}^{7, 9\cdot 2^{2}}(\mathbb F_2, \mathbb F_2)$ and $h_6D_2\in {\rm Ext}_{\mathscr A}^{7, 2^{7}}(\mathbb F_2, \mathbb F_2)$ are not in the image of the transfer, and that every indecomposable element in the $Sq^{0}$-family $\{Q_2(t) = (Sq^{0})^{t}(Q_2(0))\in {\rm Ext}^{7, 2^{t+6}}_{\mathscr A}(\mathbb F_2, \mathbb F_2):\, t\geq 0\}$ belongs to the image of the transfer.